p$ his expected payoff is \begin{equation} \label{e100} \displaystyle{ \pi_1(v_2| v_2>p) = -c + \int_{u= - \infty }^{v_2- \mu } (0) f(u) du + \int_{u=v_2- \mu }^{\infty } (\mu + u -v_2 ) f(u) du. } \end{equation} Integrating over the possible values of $v_2$ yields \begin{equation} \label{e101} \displaystyle{ pi_1 = \int_{v_2=\alpha}^{p} (\mu - v_2) h(v_2) dv_2 + \int_{v_2=p}^{\beta} \left( -c + \int_{u=v_2- \mu}^{\infty } (\mu + u -v_2 ) f(u) du \right) h(v_2) dv_2 } \end{equation} If, on the other hand, $p > \mu$, then Bidder 1 is following the policy of no discovery, and his expected payoff is the first part of equation (\ref{e101}): \begin{equation} \label{e101a} \displaystyle{ \pi_1 (p>\mu) = \int_{v_2=\alpha}^{\mu} (\mu - v_2) h(v_2) dv_2. } \end{equation} \noindent {\bf Proposition:} {\it The optimal discovery level, $p^*$, rises with $c$, rising strictly if $p^* \in (\alpha, \mu)$. Bidder 1 will follow a policy of early discovery ($p^* \in [0, \alpha)$) if $c$ is low enough, late discovery ($p^* \in [ \alpha,\mu]$) for higher levels of $c$, and no discovery ($p^* \in ( \mu, \infty]$) if $c$ is sufficiently high. } \noindent {\it Proof:} Differentiating equation (\ref{e101}) with respect to $p$ yields \begin{equation} \label{e101b} \begin{array}{ll} \frac{d \pi_1 }{dp} & = \displaystyle{ (\mu - p) h(p) - \left( -c + \int_{u=p- \mu}^{\infty } (\mu + u -p ) f(u) du \right) h(p)} \\ & \\ & = \displaystyle{ \left[ c + (\mu - p) - \int_{u=p- \mu}^{\infty } (\mu -p + u ) f(u) du \right] h(p) } \\ & \\ & \displaystyle{= \left[ c + -\int_{u=-\infty}^{\infty } (\mu - p+u) f(u) du - \int_{u=p- \mu}^{\infty } (\mu -p + u ) f(u) du \right] h(p) } \\ & \\ & = \displaystyle{ \left[c + \int_{u=-\infty}^{p- \mu } (\mu -p + u ) f(u) du \right] h(p) } \\ \end{array} \end{equation} If $h(p)>0$ (true between $\alpha$ and $\mu$) and $c$ is small enough, derivative (\ref{e101b}) is negative. If $c$ is small enough, $\frac{d \pi_1 }{dp}<0$ for $p \in [\alpha, \mu]$, and the payoff rises if $p$ is reduced to below $\alpha$ -- that is, to early discovery. If $p<\alpha$, then $h(p)=0$, so further reductions are unimportant. If $c$ is greater, $\frac{d \pi_1 }{dp} >0$ at $p=\alpha$, and the optimal $p$ exceeds $\alpha$. At the optimal $p$, \begin{equation} \label{e101c} \begin{array}{ll} \frac{d^2 \pi_1 }{dp^2} &\displaystyle{ = \left( (\mu -p + [p-\mu] ) f(u) + \int_{u=-\infty}^{p- \mu } (-1) f(u) du \right) h(p) + \left[ c+ \int_{u=-\infty}^{p- \mu } (\mu -p + u ) f(u) du \right] h'(p) } \\ &\\ & = \displaystyle{ 0 - \left( \int_{u=-\infty}^{p- \mu } f(u) du \right) h(p) + (0)h'(p) <0} \\ \end{array} \end{equation} where we use the fact that $\frac{d \pi_1 }{dp}=0$ at the optimum to obtain the term $(0)h'(p)$. Since \begin{equation} \label{e101d} \begin{array}{ll} \frac{d^2 \pi_1 }{dp dc} & = (1) h(p) >0, \\ \end{array} \end{equation} the implicit function theorem tells us that $\frac{d p }{ dc} >0$ when $h(p) > 0$, i.e., the optimal discovery level rises with the discovery cost. Thus, there exist levels of $c$ such that the optimal discovery level lies within $(\alpha, \mu)$ and late discovery is optimal. As $c$ increases, the optimal discovery level exceeds $\mu$, so ``no discovery'' becomes optimal. $\blacksquare$ \bigskip If the discovery cost is low enough, early discovery is best, because the bidder averts the possibility that he might pay more than his value by winning even at the other bidder's lowest possible value. If the discovery cost is somewhat higher, it is not worth avoiding that possibility--thus, late discovery. How late depends on the size of the discovery cost, and the optimal discovery level rises smoothly with the discovery cost. If the discovery cost is too high, then no discovery becomes optimal. This model provides an interpretation for ``getting carried away''. Suppose we see a bidder winning an auction at a price higher than his initial reservation price, and he later regrets having won--- an ``unhappy victory.'' Here is the model's interpretation. At the start of the auction, $\mu$ was the most he intended to bid. The auction began, and the bidding rose to $\mu$. he reconsidered, at cost $c$, and raised his bid ceiling to $(\mu+u)$. This new ceiling exceeded $\beta$, the most other bidders would pay, so he won, at price $\beta$. After the auction is over, however, he learned $\epsilon$ and found that $\mu+u+\epsilon<\beta $. His reaction is, ``I got carried away and bid too much. I wish I'd stuck with my original ceiling of $\mu$.'' This will not happen in every auction for every bidder, but the winning bidder is likely to be one who revised his value upwards, and if the final estimate is unbiased with a symmetric error it is as likely to be too high as too low. Thus, roughly half the time the revising bidder will be willing to pay too much, and often competition will lead him to do so. Observationally, this will look like getting carried away by emotion. An empirical test would be whether bidders who revise their reservation prices upwards regret doing so {\it on average} (emotion) or not (rational value discovery). \newpage \noindent \textbf{References} Compte, Olivier \& Philippe Jehiel (2000) ``On the Virtues of the Ascending Price Auction: New Insights in the Private Value Setting,'' working paper, CERAS-ENPC, CNRS, Paris, http://www.enpc.fr/ceras/jehiel/ascend.pdf (December 2000). Compte, Olivier \& Philippe Jehiel (2004) ``Auctions and Information Acquisition: Sealed-Bid or Dynamic Formats?'' working paper, CERAS-ENPC, CNRS, Paris, http://www.enpc.fr/ceras/compte/ascendR.pdf (May 2004) Ku, Gillian, Deepak Malhotra \& J. Keith Murnighan (2005): ``Towards a Competitive Arousal Model of Decision Making: A Study of Auction Fever in Live and Internet Auctions,'' {\it Organizational Behavior and Human Decision Processes}, 26: 89-103. Rasmusen, Eric (2003) ``Strategic Implications of Uncertainty Over One's Own Private Value in Auctions,'' January 26, 2003, \\ http://rasmusen.org/papers/auction.pdf. \end{raggedright} \end{document}